Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 04

Nov 28, 2014.

1. Estimate h or n such that the composite Trapezoidal rule for ∫_{0}^{1} sin(x^{2})dx has absolute error less than 10^{−5}. Then give your numerical value I_{h} with the n you obtained (write down your answer and hand in the program).

2. Derive the closed Newton-Cotes formula on [0, 1] with four quadrature points.

3. Derive the Gaussian quadrature on [−1, 1] with two quadrature points.

4. Describe the procedure to evaluate ∫_{0}^{1} \frac{sin x}{\sqrt{x}}dx so that the Simpson’s method would have normal fourth order accuracy. Give all details and need not find the numerical value.

Numerical Analysis I, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Quiz 04

Nov 28, 2014.

1. Estimate h or n such that the composite Trapezoidal rule for ∫_{0}^{1} sin(x^{2})dx has absolute error less than 10^{−5}. Then give your numerical value I_{h} with the n you obtained (write down your answer and hand in the program).

2. Derive the closed Newton-Cotes formula on [0, 1] with four quadrature points.

3. Derive the Gaussian quadrature on [−1, 1] with two quadrature points.

4. Describe the procedure to evaluate ∫_{0}^{1} \frac{sin x}{\sqrt{x}}dx so that the Simpson’s method would have normal fourth order accuracy. Give all details and need not find the numerical value.